We build models which attempt to capture aspects of classical physics:
the models are local, deterministic, have exact conservations, and
are exactly microscopically reversible.
We implement the models as lattice gas cellular
automata, on a special purpose cellular automata machine, the
CAM8[4] (which was developed here at MIT in the course of
the past decade). The analysis of the models relies primarily on
the standard techniques of statistical mechanics, which enables the
prediction of macroscopic quantities through a statistical analysis
of the microscopic details of the system. In some cases we can also
directly take the continuum limit of the discrete, microscopic dynamics
to establish the partial differential equations obeyed in the macroscopic
limit for the system.

Two interesting systems we are concurrently studying are the Reversible Aggregation (RA) model and the three-dimensional SAME model. The first model is of cluster growth via the diffusive aggregation of particles in a closed system. The particles release latent heat into a heat bath upon aggregating, while singly connected cluster members can absorb heat and evaporate. Initially the heat bath is empty, so aggregation events happen more readily than evaporation. The heat bath quickly reaches a steady state, meaning the system now has a well defined temperature, and that the net number of aggregation and evaporation events are equal. The growth morphology of the cluster correspondingly undergoes a transition from the typical ``frost on a window pane'' pattern observed for irreversible models of diffusive cluster growth[5], to the highest entropy macrostate allowed for a connected cluster in a finite volume, a branched polymer. We have studied the thermodynamics of this model extensively, as well as quantifying the transition in the resulting growth morphology via the fractal dimension of the clusters[6].

The second model, the three-dimensional SAME model, provides a nice example of the emergence of large scale order with no explicit entropy sink (i.e., no heat bath). The model is a generalization of the dynamical Ising model[2] of binary spins on a lattice which interact with nearest neighbor sites. The system evolves so as to maximize the entropy. In the analogous two dimensional dynamics we observe this system evolve from a small block of randomness in a uniform background to a completely random and disordered state. In contrast, in three dimensions the system evolves from a small cube of randomness to a stable dynamical structure with long range order resembling a square pyramid embedded in fluctuating spherical shells. Discrete systems have a finite, but extremely large, number of distinct microscopic states. In general, reversible systems visit almost all of the states before cycling back to the initial state, thus the stable structure generated by the SAME model will persist for longer than we could ever possibly observe it. A general discussion of this model and of cycle times in discrete reversible systems appears in Ref.[1], a detailed discussion will appear in the soon to be released thesis of the first author.

[2] M. Creutz, "Deterministic Ising Dynamics",

[3]A. L. Barabasi and U. E. Stanley,

[4] N. H. Margolus, "CAM-8: a computer architecture based on cellular automata", in

[5] T. A. Witten and L. M. Sander,"Diffusion-Limited Aggregation: a kinetic critical phenomenon",

[6] R. M. D'Souza and N. H. Margolus,"Reversible aggregation in a lattice gas model using coupled diffusion fields",submitted to